Spanning k-trees and distance spectral radius in graphs

Let \(k\ge 2\) $k\ge 2$ be an integer. A tree T is called a k-tree if \(d_T(v)\le k\) $d_T(v)\le k$ for each \(v\in V(T)\) $v\in V(T)$ ; that is, the maximum degree of a k-tree is at most k. A k-tree T is a spanning k-tree if T is a spanning subgraph of a connected graph G. Let \(\lambda _1(D(G))\) ${\lambda }_1(D(G))$ denote the distance spectral radius in G, where D(G) denotes the distance matrix of G. In this paper, we verify an upper bound for \(\lambda _1(D(G))\) ${\lambda }_1(D(G))$ in a connected graph G to guarantee the existence of a spanning k-tree in G.